One of the fundamental axioms of computer science is the Church-Turing
thesis, which states that any function computable by a ``realistic''
computer can be computed by a Turing machine. An extension of the
Church-Turing thesis, the Polynomial Church-Turing thesis states:

...any reasonable attempt to model mathematically
computer algorithms and their time performance is bound to end up with
a model of computation and associated time cost that is equivalent to
Turing machines within a polynomial [15].

This says essentially that all physically realizable computing devices
are, within a polynomial factor, equivalent to one another in their
time complexity.

In the early 1980's physicist Richard Feynman observed that no
classical computer can simulate a quantum mechanical system of
particles without incurring exponential slowdown
[18]. He also suggested that a computer that
behaves in a quantum-mechanical way could potentially simulate such
systems without exponential slowdown [18]. The
possibility that a quantum computer could violate the polynomial
Church-Turning thesis made the study of quantum computation appealing,
and provided a strong incentive for studying quantum time complexity.
Many of the earliest problems and algorithms for quantum computers
were explicitly designed to show tasks that a quantum computer
performs exponentially faster than a classical Turing machine.